Transactions of the Operations Research Society of Japan Vol. 59, 2016, pp. 21 37 c J1 2 + ( 2015 3 14 ; 2015 11 6 ) 2015 J 2 3 4 5 62%, 35%, 3%( 3.4) 5 10 2 1 : 1. J1 2015 2 + [8]( ) 1993 J 2004 2 + 2005 2014 1 ( ) ( ) ( ) ( ) ( [10]) ( [19]) & J ( ) [23]. J [17] 4 NFL( ) MLB( ) NBA( ) NHL( ) 4 MLS( ) 21
22 [21] 2 (NPB) 2007 ( ) 3 ( ) J J ( ) 2 + 2 [5] WBC 2 2015 J1 2 + 3 1 4 2 1 2 + 5 2. 2015 J1 2 + 2015 J1 + [9] 18 2 1 World Baseball Classic
J1 2 + 23 17 153 306 1 1 2 3 4 1 1st 1 2nd 1 3 1st 1 2nd 1 2 1 1 1 1 1 1 2 3 1 1 1 1 3 Y1 Y2 Y3 W1 W2 1 5 (J [9] ) 1 8 1: 2015 J1 2.1. & 1 ( ) FIFA 50 (2015 2 19 ) 2
24 case # overlap(s) teams 1 null 5 2 (Y3, W1) 4 3 (Y2, W1) 4 4 (Y2, W1), (Y3, W2) 3 1: case # overlap(s) teams 5 (Y1, W1) 4 6 (Y1, W1), (Y3, W2) 3 7 (Y1, W1), (Y2, W2) 3 8 (Y1, W1), (Y1, W2) 3 2: FIFA 50 (2015 2 ) Confederation Postseason YES NO All 15 35 UEFA (Europa) 6 25 CONMEBOL (South America) 3 3 CONCACAF (North & Central America) 4 0 CAF (Africa) 2 7 AFC, OFC (Asia, Oceania) 0 0 1 [6] V [22] 2 ( ) 2 (League Conference ) 4 8 (Division ) [14, 16] MLS(Major League Soccer) 2 + [13] ( MLS Cup Supporters Shield ) bj [1] J 1
J1 2 + 25 2 3. 1 (W1 W2) 3.1. i j λ ALL 1 λ i,gf i 1 ( ) (Goals For). λ i,ga i 1 ( ) (Goals Against). λ i,gf,h λ i,ga,h i 1 λ i,gf,a λ i,ga,a X i i 1 P o(λ) λ. 3.2. 1 [2, 12]. 2 2013 J1 1 0.4 0.35 Observed Poisson 0.3 Frequency 0.25 0.2 0.15 0.1 0.05 0 0 1 2 3 4 5 6 7 8 9 10 Goals 2: 1 J1 ( 3 1 0) 1
26 [18] [4] Maher [12] ( ) M1: X i P o(λ ALL ) p(x) = e λ λx ALL ALL, x = 0, 1,. (3.1) x! P (X i = x, X j = y) = p(x)p(y). (3.2) M2: X i P o(λ i,gf ) p i (x) = e λ i,gf λx i,gf x!, x = 0, 1,. (3.3) P (X i = x, X j = y) = p i (x)p j (y). (3.4) ( ) λi,gf + λ j,ga M3: i, j X i P o 2 p i,j (x) = e µ µx i,j i,j x!, x = 0, 1,, µ i,j = λ i,gf + λ j,ga. (3.5) 2 P (X i = x, X j = y) = p i,j (x)p j,i (y). (3.6) ( ) λi,gf,h + λ j,ga,a M4: i j X i P o 2 p i,j (x) = e µ µx i,j i,j x!, x = 0, 1,, µ i,j = λ i,gf,h + λ j,ga,a. (3.7) 2 P (X i = x, X j = y) = p i,j (x)p j,i (y). (3.8)
J1 2 + 27 M5: i j i g (λ i,gf, λ j,ga, g) (λ i,gf, λ j,ga ) ( ) µ i,j µ(λ i,gf, λ j,ga ) = a 1 λ i,gf + a 2 λ j,ga + a 3 (3.9) i, j X i P o (µ i,j ) p i,j (x) = e µ µx i,j i,j, x = 0, 1,. (3.10) x! 4. 4.1. P (X i = x, X j = y) = p i,j (x)p j,i (y). (3.11) λ i,gf, λ i,ga J M5 5 (2010 2014) 3 M5 (λ j,ga ) (λ i,gf ) 0.4 0.4 25 30 ( 542) 3062 2968 1.3690 1.3359 ( 3) 3: (2010-2014 ) Year 2014 2013 2012 2011 2010 Average goals 1.265 1.436 1.376 1.420 1.328 µ i,j = 0.8301λ i,gf + 0.8192λ j,ga 0.8973 (4.1) r 2 = 0.8721
28 3: 1. 18 2. 3. 4. 3 10 2013 4 5 4: Teams M1 M2 M3 M4 M5 case # 3 0.4320 0.5912 0.5196 0.5676 0.6173 4, 6, 7, 8 4 0.4876 0.3740 0.4282 0.3916 0.3519 2, 3, 5 5 0.0804 0.0348 0.0522 0.0408 0.0308 1 mean 3.6485 3.4466 3.5327 3.4372 3.4134 4.2. 4 5 10 30 1 M1 (case 1) 8% J 5
J1 2 + 29 5: Teams M1 M2 M3 M4 M5 case # 5 0.0804 0.0348 0.0522 0.0408 0.0308 1 4 0.0754 0.0496 0.0606 0.0527 0.0459 2 4 0.1205 0.0865 0.1016 0.0898 0.0809 3 3 0.0530 0.0549 0.0539 0.0513 0.0550 4 4 0.2917 0.2379 0.2660 0.2491 0.2251 5 3 0.1168 0.1399 0.1328 0.1353 0.1448 6 3 0.2063 0.2766 0.2456 0.2639 0.2864 7 3 0.0559 0.1198 0.0873 0.1172 0.1311 8 M1 M4 4 2013 (63) M2 M4 3.5 4 x 104 3 M2 M3 M4 Frequency 2.5 2 1.5 1 0.5 0 0 20 40 60 80 100 Points 4: M2 M4 ( 48.705, 54.686, 48.797 10 15 ) M5 6 2013 (Pts) (Mean) (Err) (Std) (Err/Std) 18 16 µ ± 1.2σ 2 M1 M4 2010 2014 5 15 (18 ) ± 5 M5
30 6: M5 ( ) Standing Pts Mean Err Std Err/Std 1 63 60.86 2.14 7.32 0.292 2 62 58.22 3.78 7.37 0.512 3 60 54.64 5.36 7.67 0.698 4 59 59.92 0.92 7.39 0.124 5 59 51.28 7.72 7.63 1.011 6 58 52.24 5.76 7.71 0.747 7 55 50.41 4.59 7.53 0.609 8 54 54.86 0.86 7.63 0.112 9 50 41.67 8.33 7.47 1.115 10 48 45.16 2.84 7.63 0.372 11 47 46.12 0.88 7.52 0.117 12 46 41.85 4.15 7.55 0.549 13 45 48.65 3.65 7.41 0.492 14 45 44.90 0.10 7.47 0.013 15 37 39.43 2.43 7.10 0.342 16 25 30.72 5.72 6.90 0.828 17 23 37.37 14.37 7.22 1.990 18 14 26.62 12.62 6.55 1.926 M5 5 0.0326 5.1129 5 χ 2 3 χ 2 = 3.7177 p = 0.2936 > 0.05 4.2.1. J 1993 2014 1 (1996, 2005-2014) 1st 2nd 7 2004 10 16 7: 1993-2014 Teams 3 4 5 Frequency 16/22 (73%) 5/22 (23%) 1/22 (4%) χ 2 2 χ 2 = 1.555 p = 0.4595 > 0.05
J1 2 + 31 1 Cumulative distribution function 0.8 0.6 0.4 0.2 Points difference Approx. cdf 0 20 10 0 10 20 Points difference between real and simulation 5: 4.3. 1 2 3 5 3 4 ( ) [7, 15] 2015 J ( ) 18 2 1 17 153 306 3 (Y1 Y3) Y1 ( 6) 4 3 (W1) 4 (W2) W2 1
32 6: W1 Y2 Y3 W1 1 W2 1 ( 7(a)) W1 Y1 W2 1 Y2 Y3 1 ( 7(b)) 7: 4 2 (W1, W2) 1 1 Y2 W2 Y3 W1 4 Y1 ( 8) 8: 4.4. 2 + 2015 J1 1
J1 2 + 33 1 (4 ) 1 [9] 2 3 3 J ( ) [23] 2 1 2 1 5 2013 9 J [7, 15] 2 2 3 1982 [20] 2 2 1986 2 4 [3] ( ) 1 [17, 67 ] 1 5 1 5 2013 9 J1 [11] 2 1 2 34 1
34 34 1 5 1 2 2 1 3 ( ) [17, 49 ] 2 2 3 5 J 5 5 5 1 1 4 3 4 1 3 4 1 1 J 5 3 ( )2 J 5. 2015 J 2 3 1 3 4 1 2 (1 ) 1 +
J1 2 + 35 [1] bj : 2014-2015, http://www.bj-league.com/championship. php (2015 2 27 ). [2] J.S. Croucher: Using Statistics to Predict Scores in English Premier League Soccer. In S. Butenko, J. Gil-Lafuente, and P.M. Pardalos (eds.): Economics, Management and Optimization in Sports (Springer Berlin Heidelberg, 2004), 43 57. [3] R. Gilmour: Badminton match-fixing scandal: how and why the four pairs were disqualified from the london 2012 olympics, http://www.telegraph.co.uk/sport/olympics/ badminton/9444025/badminton-match-fixing-scandal-how-and-why-the-four-pairswere-disqualified-from-the-london-2012-olympics.html (2015 3 2 ). [4] J. Greenhough, P.C. Birch, S.C. Chapman, and G. Rowlands: Football goal distributions and extremal statistics. Physica A: Statistical Mechanics and its Applications, 316 (2002), 615 624 [5],, : (WBC)., 57 (2012), 629 638. [6] : 39, http://www.top-league.jp/about/kiyaku/ 2013/a/03.html#39. (2015 2 25 ). [7] J : 10 30, http://www.j-league.or.jp/release/000/00005433.html (2014 11 28 ). [8] J : 2015 ( ), http://www.j-league.or.jp/release/000/00005661.html (2014 11 28 ). [9] J : 2015 J1, http://www.jleague.jp/ aboutj/construction/ (2015 2 15 ). [10] Jupiler Pro League: Formule de championnat. http://www.sport.be/fr/jupilerpro league/competitieformule/ (in French) (2014 11 28 ). [11] : J1 ( ), http://www. frontale.co.jp/info/2013/0922_1.html (2015 2 27 ). [12] M.J. Maher: Modelling association football scores. Statistica Neerlandica, 36 (1982), 109 118. [13] Major League Soccer: Competition rules and regulations, http://pressbox.mlssoccer. com/content/competition-rules-and-regulations. (2015 2 25 ). [14] Major League Baseball: World series history, http://m.mlb.com/postseason/history (2015 2 25 ). [15] MSN : 2 1, http://sankei. jp.msn.com/sports/news/131030/scr13103019190009-n1.htm. (2014 11 28 ). [16] National Football League: History 1941 1950, http://www.nfl.com/history/chronology /1941-1950 (2015 2 25 ). [17], ( ), ( ): J (, 2014). [18] C. Reep, R. Pollard, and B. Benjamin: Skill and chance in ball games. Journal of the Royal Statistical Society. Series A (General), 134 (1971), 623 629.
36 [19] Scottish Professional Football League: The rules of the Scottish Professional Football League, http://spfl.co.uk/docs/067_324 therulesofthescottishprofessionalfootball leagueasat11september2014_1411980004.pdf (2014 11 28 ). [20] R. Smyth: World cup: 25 stunning moments... no3: West germany 1-0 austria in 1982, http://www.theguardian.com/football/blog/2014/feb/25/world-cup-25-stunningmoments-no3-germany-austria-1982-rob-smyth (2015 3 3 ). [21] : 2 ( issue65 ), http:// www.footballchannel.jp/2013/11/15/post12391/ (2014 11 28 ). [22] V : 2014/15 V, http://www. vleague.or.jp/news_topics2/article/id=12463 (2014 12 19 ). [23] ( ): J 2, http://number.bunshun.jp/articles/-/724962 (2014 11 28 ). 468-8502 1-501 E-mail: konaka@meijo-u.ac.jp
J1 2 + 37 ABSTRACT STATISTICAL ANALYSIS OF TWO-STAGE AND POSTSEASON FORMAT OF J1 FOOTBALL LEAGUE Takeshi IZUMI Eiji KONAKA Meijo University In this paper, the new postseason format from 2015 season of J1 League, the top division of professional football in Japan, is simulated and analyzed statistically. Official regulation defines that the new postseason from the 2015 season consists of five teams selected by different principles the top three teams by total season points, and the winning teams of the 1st and 2nd half of the season. The simulation results concludes that there are overlaps within these five teams very frequently, therefore the postseason tournament will be held with 3, 4, and 5 teams (i.e., without any overlaps) with probability 62%, 35%, and 3%. The result is obtained by using numerical simulation of 10 5 seasons. The goal scoring model is based on the results in 5 years (from 2010 to 2014 seasons), and is constructed by regression analysis technique. This result clarifies that the new postseason format is inherently one-stage, NOT two-stage as officially defined, format because the selection condition for the postseason tournament is not designed correctly.